On 2d Conformal Field Theories with Two Characters

Rational CFT’s are classified by an integer $\mathrm{\ell }$, the number of zeroes of the Wronskian of their characters in moduli space. For $\mathrm{\ell }=0$ they satisfy non-singular modular-invariant differential equations, while for $\mathrm{\ell }>0$ the corresponding equations have singularities. We survey CFT’s with two characters and $\mathrm{\ell }=0,2,3,4$ and verify the consistency, at the level of characters, of some candidate theories with $\mathrm{\ell }\mathrm{\ne }0$. For $\mathrm{\ell }=2$ there are seven consistents sets of characters. We identify specific combinations of level-1 current algebras that are potential symmetries of the corresponding CFT’s.